Abstract:
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Most multiple testing procedures assume exact knowledge of all p-values, for instance the ones of Holm, Sidak, Hochberg or Benjamini-Hochberg. We consider multiple testing under the assumption that the p-values are not available and thus have to be approximated using Monte-Carlo simulations. We are interested in obtaining the same rejections and non-rejections as the ones obtained if all p-values had been available. This talk introduces a framework for this common scenario by providing a generic algorithm for a general testing procedure. We establish conditions which guarantee that the rejections and non-rejections obtained through Monte-Carlo simulations are identical to the ones obtained with the unknown p-values. Our framework is applicable to a general class of step-up and step-down procedures. More importantly, we show how to use our framework to improve established methods in such a way as to yield theoretical guarantees on their results. These modifications can easily be implemented in practice and lead to a particular way of reporting multiple testing results as three sets together with an error bound on their correctness, demonstrated exemplarily on a biological dataset.
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